96,241 research outputs found
Optimal Navigation Functions for Nonlinear Stochastic Systems
This paper presents a new methodology to craft navigation functions for
nonlinear systems with stochastic uncertainty. The method relies on the
transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear
partial differential equation. This approach allows for optimality criteria to
be incorporated into the navigation function, and generalizes several existing
results in navigation functions. It is shown that the HJB and that existing
navigation functions in the literature sit on ends of a spectrum of
optimization problems, upon which tradeoffs may be made in problem complexity.
In particular, it is shown that under certain criteria the optimal navigation
function is related to Laplace's equation, previously used in the literature,
through an exponential transform. Further, analytical solutions to the HJB are
available in simplified domains, yielding guidance towards optimality for
approximation schemes. Examples are used to illustrate the role that noise, and
optimality can potentially play in navigation system design.Comment: Accepted to IROS 2014. 8 Page
Using hybrid GPU/CPU kernel splitting to accelerate spherical convolutions
We present a general method for accelerating by more than an order of
magnitude the convolution of pixelated functions on the sphere with a
radially-symmetric kernel. Our method splits the kernel into a compact
real-space component and a compact spherical harmonic space component. These
components can then be convolved in parallel using an inexpensive commodity GPU
and a CPU. We provide models for the computational cost of both real-space and
Fourier space convolutions and an estimate for the approximation error. Using
these models we can determine the optimum split that minimizes the wall clock
time for the convolution while satisfying the desired error bounds. We apply
this technique to the problem of simulating a cosmic microwave background (CMB)
anisotropy sky map at the resolution typical of the high resolution maps
produced by the Planck mission. For the main Planck CMB science channels we
achieve a speedup of over a factor of ten, assuming an acceptable fractional
rms error of order 1.e-5 in the power spectrum of the output map.Comment: 9 pages, 11 figures, 1 table, accepted by Astronomy & Computing w/
minor revisions. arXiv admin note: substantial text overlap with
arXiv:1211.355
Gap Processing for Adaptive Maximal Poisson-Disk Sampling
In this paper, we study the generation of maximal Poisson-disk sets with
varying radii. First, we present a geometric analysis of gaps in such disk
sets. This analysis is the basis for maximal and adaptive sampling in Euclidean
space and on manifolds. Second, we propose efficient algorithms and data
structures to detect gaps and update gaps when disks are inserted, deleted,
moved, or have their radius changed. We build on the concepts of the regular
triangulation and the power diagram. Third, we will show how our analysis can
make a contribution to the state-of-the-art in surface remeshing.Comment: 16 pages. ACM Transactions on Graphics, 201
Approximate Computation and Implicit Regularization for Very Large-scale Data Analysis
Database theory and database practice are typically the domain of computer
scientists who adopt what may be termed an algorithmic perspective on their
data. This perspective is very different than the more statistical perspective
adopted by statisticians, scientific computers, machine learners, and other who
work on what may be broadly termed statistical data analysis. In this article,
I will address fundamental aspects of this algorithmic-statistical disconnect,
with an eye to bridging the gap between these two very different approaches. A
concept that lies at the heart of this disconnect is that of statistical
regularization, a notion that has to do with how robust is the output of an
algorithm to the noise properties of the input data. Although it is nearly
completely absent from computer science, which historically has taken the input
data as given and modeled algorithms discretely, regularization in one form or
another is central to nearly every application domain that applies algorithms
to noisy data. By using several case studies, I will illustrate, both
theoretically and empirically, the nonobvious fact that approximate
computation, in and of itself, can implicitly lead to statistical
regularization. This and other recent work suggests that, by exploiting in a
more principled way the statistical properties implicit in worst-case
algorithms, one can in many cases satisfy the bicriteria of having algorithms
that are scalable to very large-scale databases and that also have good
inferential or predictive properties.Comment: To appear in the Proceedings of the 2012 ACM Symposium on Principles
of Database Systems (PODS 2012
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